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E39 (1997 - 2003) The BMW 5-Series (E39 chassis) was introduced in the United States as a 1997 model year car and lasted until the 2004 when the E60 chassis was released. The United States saw several variations including the 525i, 528i, 530i and 540i. -- View the E39 Wiki

ok this happens often when wet outside what i do is manually shift by going from drive to 3rd gear and back up this shifts it then i would check the abs light it could be water just got on it or that it shorted but that would be my advice i have no idea if this is bad for trans ive done it a lot and my trans is still rocking again just my experience

Wheel speed sensors... rear drivers side wheel speed sensor will cause speedometer issues and front passenger wheel speed sensor affects the shifting . This was the first issue I dealt with when I bought my 97 528i. I had same lights lit and same symptom.